The Championship club will pay an undisclosed compensation fee for the 21-year-old, who made 42 appearances for the Dons last season.
Chicksen has signed a three-year contract and will join his new team-mates on the pre-season trip to Spain.
His arrival follows this week's permanent deal for former England centre-half Matthew Upson.
Albion's head of football operations David Burke told the club website: "We are delighted that Adam has joined us. Our scouting department have been aware of him and impressed with his progress at MK Dons.
"He has all the attributes that we are looking for and already has a good knowledge of our playing style, having come through the ranks at MK Dons.
"He has great potential to establish himself in the Championship, and we are delighted that he has chosen Brighton & Hove Albion."